LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 41
The continuum Euler-Arnold equations for the motion of an N-dimensional rigid body
are given by
^ £ - [ M , f t ] . , (2.142)
where Q £ o(N) is defined through
M= JQ + QJ . (2.143)
These equations are the continuum limit of the discrete Euler-Arnold equation (1.12) (see
[MV]) and, as noted above, they have the same integrals. Question: is equation (1.12) the
time-r map for (2.142) for some r 0?
The answer is "no" by the following argument. The interpolating flow for (1.12) is
^ = [(w_ log M(tt •) )(0), M] . (2.124)
Evaluating (TT- log M(t, •) )(0) for small M, we find after a straightforward calculation that
[(7r-logM(t,.)(0),A/] = -[ft,M] + K(M) + 0(M6) , (2.144)
where
K{M)
v fir ( f r 1 ,** l x3 w i i ds \ A" dX ( 2 - 1 4 5 )
T-ttJ-irJc s- \_xi s- \*xi 2 T T Z
/
( 1 -
A2)3
2m
and so (2.124) becomes
^ = [ M , f i ] + / « M ) + 0 ( M 6 ) ,
which agrees with (2.142) to third order. However a simple computation shows that K(M)
is not identically zero for all J and M, and hence (2.142) cannot interpolate (1.12) for any
r 0.
As a final remark we note that in the continuum limit, M = em(et), e | 0, equation
(2.124) reduces in scaled time T = et to the equation
( m -f
fiJ2)
= [m +
f.iJ2,
Q, -f fiJ] ,
m = Jft-f ftJ ,
which is Manakov's Lax-pair form for the Euler-Arnold equations (see [Man]).
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